One of the most common uses for dummy variables is for fixed effects. A common use case for fixed effects is when you are working with a time-by-unit panel. For example: a state-by-year panel.
By including state dummies, or fixed effects, we are removing / controlling for any and all constant (time-invariant) factors. As an example, if the outcome variable in our state-by-year panel is housing prices, we can reasonably expect California to have high prices and West Virginia to have low prices. Why? For a variety of reasons: weather and other local amenities, economic factors, etc. To make the states more comparable to one another, we can remove the effect of the idiosyncratic differences intrinsic to each state.
We can also include year fixed effects, which control for any and all factors that impact each state at the same time in the same way. For example, the 2007 financial crisis reduced housing prices across the country. When we include dummies for each year (except one, the reference group), we can remove the influence of these events on our outcome.
When using fixed effects, we write our equations a bit differently. If we have a state-by-year panel, you will see equations written like the following:
where \(\beta_s\) represents state (\(s\)) fixed effects and \(\beta_t\) represents time (\(t\)) fixed effects. Notice how we do not include \(\beta_0\) (the intercept). This is because the fixed effects take the place of the intercept in practice.
When we estimate fixed effects with lm(), R still reports an intercept. That is because R drops one dummy from each set of fixed effects (the reference groups, like we saw in Module 6) to avoid perfect multicollinearity. The intercept is then the expected outcome for the reference state in the reference year.
Income and Housing
To demonstrate fixed effects, we are going to compare per-capita personal income and housing price indices for different counties/cities in Hampton Roads over time. Let’s take a look at the data files we have.
First, we have the FIPS codes for each of the areas of interest. FIPS codes are unique identifiers for counties/cities in the United States. The first two digits denote the state and the final three numbers denote the county or city within that state.
Code
hr <-read.csv("https://alexcardazzi.github.io/econ311/data/hr_fips.csv")hr # hr is "hampton roads"
Output
name fips
1 Gloucester County 51073
2 Isle of Wight County 51093
3 James City County + Williamsburg City 51931
4 Mathews County 51115
5 York County 51199
6 Chesapeake City 51550
7 Hampton City 51650
8 Newport News City 51700
9 Norfolk City 51710
10 Portsmouth City 51740
11 Suffolk City 51800
12 Virginia Beach City 51810
The other two datasets we are going to use contain per-capita personal income and house price index values. Both contain observations by FIPS code and year.
Code
inc <-read.csv("https://alexcardazzi.github.io/econ311/data/pcpi.csv")head(inc, 12) # inc is "personal income"
Output
series_id value fips year
1 Per Capita Personal Income 3696 51073 1969
2 Per Capita Personal Income 3916 51073 1970
3 Per Capita Personal Income 4278 51073 1971
4 Per Capita Personal Income 4812 51073 1972
5 Per Capita Personal Income 5115 51073 1973
6 Per Capita Personal Income 5329 51073 1974
7 Per Capita Personal Income 5815 51073 1975
8 Per Capita Personal Income 6493 51073 1976
9 Per Capita Personal Income 6986 51073 1977
10 Per Capita Personal Income 7819 51073 1978
11 Per Capita Personal Income 8297 51073 1979
12 Per Capita Personal Income 9478 51073 1980
Code
hp <-read.csv("https://alexcardazzi.github.io/econ311/data/athpi.csv")head(hp, 12) # hp is "housing price"
Output
series_id value fips year
1 All-Transactions House Price Index 83.51 51073 1993
2 All-Transactions House Price Index 83.21 51073 1994
3 All-Transactions House Price Index 85.19 51073 1995
4 All-Transactions House Price Index 86.72 51073 1996
5 All-Transactions House Price Index 88.11 51073 1997
6 All-Transactions House Price Index 92.83 51073 1998
7 All-Transactions House Price Index 95.75 51073 1999
8 All-Transactions House Price Index 100.00 51073 2000
9 All-Transactions House Price Index 104.24 51073 2001
10 All-Transactions House Price Index 109.65 51073 2002
11 All-Transactions House Price Index 115.22 51073 2003
12 All-Transactions House Price Index 131.49 51073 2004
We will need to “merge” these three data files in order to run our regression. In other words, we need to pull information from one file and place it in another file.
Merging Datasets
To begin merging, we should check out which data set has the most “coverage”. For example, the inc dataset contains the range of years {1969, 2021} whereas hp contains {1975, 2022}. This can also be seen graphically:
Code
par(mfrow =c(1, 2))plot(table(inc$year), las =1,main ="Personal Income",xlab ="Year", ylab ="Frequency")plot(table(hp$year), las =1,main ="Housing Price Index",xlab ="Year", ylab ="Frequency")par(mfrow =c(1, 1))
Plot
These figures suggest we should merge intoinc, or else we would potentially be dropping data. It’s always a good idea to have too much than too little.
Next, we are going to merge in the names from hr into inc so we can have a better way to decipher which FIPS code is which county/city. To do this, we are going to match the entries in fips in inc to the entries in fips in hr. We are going to use match() to help us with this.
match() accepts two arguments. The first is the values to be matched, and the second is the values to match against. The resulting vector tells us the location of each element of the first vector (inc$fips) in the second vector (hr$fips).
As a simple example, consider the following. I have a short vector of colleges/universities (v1). We have a second data source (ex) that contains the state these institutions are located. Use the match function below to pull the state for each institution in the first vector from the second data source. The result should be "VA" "WV" "NJ".
Now let’s use the same code to pull the FIPS names from hr.
Code
m <-match(inc$fips, hr$fips)inc$name <- hr$name[m]View(inc[,c("value", "fips", "year", "name")])
Output
It worked! We have successfully pulled over the county name for each FIPS code.
Next, we need to pull in the housing price index data from hp. We can try a similar process:
Code
m <-match(inc$fips, hp$fips)inc$hp <- hp$value[m]View(inc[,c("value", "fips", "year", "hp")])
Output
While we were able to pull a number from the dataset, the code did not work as we intended. There should be variation in hp across years. When using match(), R will always return the first instance of a match, but that is not necessarily what we want. Since we did not specify anything about years, R pulled the first value for each FIPS which is not what we’re looking for.
Merging by Two Variables
We need to create a variable that contains information about FIPS and year. To do this, we are going to use paste(). This function takes two or more variables and “pastes” them together into one. As an example: paste(c("ODU", "WVU"), c("Monarchs", "Mountaineers")): ODU Monarchs, WVU Mountaineers.
Code
m <-match(paste(inc$fips, inc$year),paste(hp$fips, hp$year))inc$hp <- hp$value[m]View(inc[,c("value", "fips", "year", "hp")])
Output
You might notice that there is quite a bit of missing data in the hp column of inc. That is because, if you recall, we have more personal income data than housing price index data. Now that we have the two variables lined up in the same data frame, let’s explore their relationship a bit. First, let’s plot the two variables in a scatterplot:
Code
inc$value000 <- inc$value/1000plot(inc$value000, inc$hp, las =1,xlab ="Personal Income (in Thousands)", pch =19,ylab ="Housing Price Index (100 is 2000 $'s)",col = scales::alpha("mediumseagreen", 0.4))
Plot
Estimating Fixed Effects
There appears to be a pretty strong, positive relationship between these variables. Now let’s take a look at what some regressions tell us about the strength of the relationships. First, we are going to estimate a model where housing price index is the outcome and personal income is the explanatory variable. Next, we are going to include FIPS and year fixed effects.
We have already investigated what happens when we include control variables into our regressions: their effects get removed from both the outcome variable and the other explanatory variables. In this case, it is likely that time is an important third variable that drives both housing prices and personal income. Let’s revisit the coefficient bias table from Module 5.3. Since time likely has a positive effect on both housing prices and personal income, the bias is likely positive. In other words, by not accounting for time, the coefficient magnitudes are being artificially pushed in positive direction.
lm() can estimate all of these dummies, but with hundreds of counties or states it gets slow and messy. In ECON 400, you will use feols() from the fixest package, which handles fixed effects for you with a | in the formula.
FWL Theorem
Let’s take a look at what happens when we partial out the effect of our “fixed effects”.
We are going to use a trick where we run two regressions with only these fixed effects. One of these regressions will have housing price as the outcome variable and the other regression will have personal income as the outcome. After each estimation, we are going to collect the residuals, which is the variation left over after having removed the influence of the fixed effects. Then, we are going to plot the two sets of residuals against each other and run a third regression between them.
Code
# First two regressions:r_hp <-lm(hp ~as.factor(fips) +as.factor(year), inc, subset =!is.na(inc$hp))r_inc <-lm(value000 ~as.factor(fips) +as.factor(year), inc, subset =!is.na(inc$hp))plot(r_inc$residuals, r_hp$residuals, las =1)# Third regression, using only the residuals:r_resid <-lm(r_hp$residuals ~ r_inc$residuals)coef(r_resid)[2]
Output
r_inc$residuals
0.4945878
Plot
The coefficient from the fixed effects model matches the coefficient from this funky residual model! Taking residuals from two equations like this is called the Frisch–Waugh–Lovell theorem.
What is left over in those residuals is called within variation: how much a county’s income and housing prices differ from that same county’s usual levels, in a year compared to other years. Fixed effects throw away all of the between variation (high-income counties versus low-income counties, expensive years versus cheap years), so the coefficient only uses the within-unit variation. This is why we say fixed effects control for anything that is constant within a county or common to all counties in a year.
The point you should take away is that often times there are unobserved idiosyncratic factors that need to be controlled for. As an example, our coefficient estimate shrunk from 3.7 to 0.49 after we included fixed effects, or by a factor of about 7.5.
Reporting Coefficients
When reporting regression models, people are usually uninterested in fixed effects. Rather, people like to include rows in their tables that indicate which fixed effects are included in which models. As an example, let’s consider the housing price and personal income regressions.
Code
library("modelsummary")options("modelsummary_factory_default"="kableExtra")fe1 <-lm(hp ~ value000, inc)fe2 <-lm(hp ~ value000 +as.factor(fips) +as.factor(year), inc)regz <-list(`Housing Price Index`= fe1,`Housing Price Index`= fe2)coefz <-c("value000"="Personal Income (in Thousands)")gofz <-c("nobs", "r.squared")rowz <-as.data.frame(matrix(c('County Fixed Effects', "No", "Yes","Year Fixed Effects", "No", "Yes"),byrow = T, nrow =2)) # 2 rowsattr(rowz, 'position') <-c(3, 4) # rows 3 and 4# since rows 1 and 2 are coefficient and standard errormodelsummary(regz,title ="Effect of Personal Income on Housing Price Index",estimate ="{estimate}{stars}",coef_map = coefz,gof_map = gofz,add_rows = rowz)
